Overdetermined system of equations

In case of serious overlappings, multivariate techniques (see Sect. 6.4) are used and p ) > n sensors (measuring points zjt) are measured for n components. From this an overdetermined systems of equations results and, therefore, non-squared sensitivity matrixes. Then the total multicomponent sensitivity is given by... 

We will begin our discussion by demonstrating that, for a non-overdetermined system of equations, the algebraic approach and the least-square approach provide the same solution. We will then extend the discussion to the case of an overdetermined system of equations. 

The least squares method is the best possible solution to an overdetermined system of equations, where we have more equations than unknowns. This is a very common situation in problems, in which, for example, we have a set of N measurements which we want to fit to an equation with M parameters (M < N). For example if we measure temperatures as a function of time... 

For a system with N atoms, there are N(N — I)/2 interatomic distances and only 3N — 6 independent degrees of freedom (the x, y, z of atoms). Therefore, equation 25 is an overdetermined systems of equations. In the LST approach equation 25... 

The usual way to solve (1) is to multiply the overdetermined system of equations... 

In order to eliminate these disadvantages recent attention has been paid in the literature to the immediate solution of the overdetermined system of equations (2). The reason is that, theoretically in terms of the solution s error, the methods for the solution of systems of linear equations based on orthogonal decomposition are considered more accurate than the methods based on triangular decomposition (Wilkinson 1965). 

Based on previous considerations the following may be concluded. Problem (1) can be solved using two approaches the method of normal equations, and immediated solution of the overdetermined system of equations (2) using the methods of orthogonal decomposition. 

If the source fingerprints, for each of n sources are known and the number of sources is less than or equal to the number of measured species (n < m), an estimate for the solution to the system of equations (3) can be obtained. If m > n, then the set of equations is overdetermined, and least-squares or linear programming techniques are used to solve for L. This is the basis of the chemical mass balance (CMB) method (20,21). If each source emits a particular species unique to it, then a very simple tracer technique can be used (5). Examples of commonly used tracers are lead and bromine from mobile sources, nickel from fuel oil, and sodium from sea salt. The condition that each source have a unique tracer species is not often met in practice. 

Overdetermination of the system of equations is at the heart of regression analysis, that is one determines more than the absolute minimum of two coordinate pairs (xj/yi) and xzjyz) necessary to calculate a and b by classical algebra. The unknown coefficients are then estimated by invoking a further model. Just as with the univariate data treated in Chapter 1, the least-squares model is chosen, which yields an unbiased best-fit line subject to the restriction ... 

Thus, we have i = 0 at the equilibrium. Equation (28) together with the linear balance Equation (27b) form an overdetermined system of As we... 

One more important property of the self-dual Yang-Mills equations is that they are equivalent to the compatibility conditions of some overdetermined system of linear partial differential equations [11,12]. In other words, the selfdual Yang-Mills equations admit the Lax representation and, in this sense, are integrable. For this very reason it is possible to reduce Eq. (2) to the widely studied solitonic equations, such as the Euler-Amold, Burgers, and Devy-Stuardson equations [13,14] and Liouville and sine-Gordon equations [15] by use of the symmetry reduction method. 

They are in line with the traditional Lie approach to the reduction of partial differential equations, since they exploit symmetry properties of the equation under study in order to construct its invariant solutions. And again, any deviation from the standard Lie approach requires solving overdetermined system of nonlinear determining equations. A more profound analysis of similarities and differences between these approaches can be found elsewhere [33,56,64]. 

Consequently, to describe all the ansatzes of the form (53),(54) reducing the Yang-Mills equations to a system of ordinary differential equations, one has to construct the general solution of the overdetermined system of partial differential equations (54),(86). Let us emphasize that system (54),(86) is compatible since the ansatzes for the Yang-Mills field ( ) invariant under the three-parameter subgroups of the Poincare group satisfy equations (54),(86) with some specific choice of the functions F, F2, , 7Mv, [35]. 

The symbol = is employed, because, in general, the system of equations is overdetermined and an equality is not possible for all of the data. 

More generally, the reaction field factors may either be determined numerically, since they appear in an overdetermined system of linear equations,23 or they may be computed analytically for certain idealized cavities (e.g., sphere and ellipsoid).30 66,213,214 Efficient optimization of solvated geometries motivates the latter approach,2i3,235-237 but the formalism has also been ap-... 

To complete our derivation of the leading-order equations for flow in the thin gap, we must specify the characteristic pressure n. It is evident in examining (5 23) (5 -25) that one of the pressure-gradient terms must be retained in the limit e 0, as the system of equations would otherwise be overdetermined. Thus, either = pH or n = p.ii/e2. Either choice gives a set of three equations for the three unknowns uf uf and p(0>. However, only the equations corresponding to n = fiQ/s2 yield solutions that can satisfy all of the boundary conditions (5-20). In particular, if n = p.i2 it can be shown that the condition uf) = 0, at y = //, can be satisfied only if dh/d() = 0 so that the cylinders are concentric and uf = 0 everywhere. Thus,... 

Generalized inverse (pseudoinverse) To obtain an approximate solntion of an overdetermined system of linear eqnations, i.e., when the number of equations is greater than the number of unknowns (m > n), a vector x is sought to minimize the square of residuals, r r, where... 

The number of observations is always much greater than the minimum necessary to compute the unknowns. Because of this, relations between unknown values and observations lead to an overdetermined system of non-linear equations ... 

It is not clear, from a practical point of view if there is an advantage in using the methods based on orthogonal decomposition for the immediate solutionof the overdetermined system of euqations (2) over the method of normal equations, so it is necessary to compare experimentally all these methods using a test stream formed from a number of large practical problems. 

Duff I.S., Reid J.K. (1976) A Comparison of some Methods for the Solution of Sparse Overdetermined Systems of Linear Equations. J. Inst. Math. Appl. Vol. 17 pp 267-280. 

Equation (4.122) can be used to determine the value of Cs. However, because this is a tensor equation, the system of equations is overdetermined. Lilly [18] proposed an expression for Cs that best suits the system given by (4.122) by minimizing the least squares error, obtaining the equation... 

Least Square FM through Solving an Overdetermined System of Linear Equations. When the force field is dependent linearly with the fitting parameters then it is possible to execute least square fit in equation 17 through the solution of an over-determined system of linear equations. A linear... 

The situation is similar to that in the degenerate perturbation theory. If we divide each equation by ci, we have only two unknown variables, c2/ci and ca/ci. If W is arbitrary, the system of equations is therefore overdetermined, with three equations for two variables. In order for a nontrivial solution to exist, a condition must be satisfied that makes the equations equivalent to two independent equations. This condition is that the determinant of the matrix of the coefficients must vanish. Determinants are discussed in Appendix B. We write the condition... 

Equation 8c is a linear system of equations that is overdetermined with three-component seismic waveform data, even with low-pass filters applied. Writing this in terms of a system of linear equations, the following is obtained ... 

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